This invention concerns the assessment of welding. In particular it concerns an apparatus and a process for determining whether a fault has occurred in a welding process, while the process is under way. The invention is applicable to gas-metal arc welding, tungsten-inert gas welding, pulsed welding, resistance welding, submerged arc welding and to other welding and cutting processes where there is an arc plasma.
The study of welding and cutting arc phenomena involves observation of both voltage and current signals having periods of milliseconds to seconds, or even micro-seconds. One way of monitoring these signals involves the use of high speed photography, and another is the use of oscillograms. The limitations inherent in the observation techniques and the difficulties in analysing the resulting data, make it difficult to provide a weld quality measurement in real time.
A Single Welding Signature
The invention is an apparatus for on line welding assessment, comprising:
first sampling means to sample the welding voltage or current to provide a sequence of values for a first signal,
second sampling means to sample the welding current or voltage to provide a sequence of a values of a second signal,
a signal generating means to generate one or more sequences of values for one or more artificial third signals from the first signal and second signals where the artificial signals depend upon values of the first and second signals through generalised discrete point convolution operations,
tripling means to identify corresponding values of the first, second and third signals, and
collection means to collect triplets of values which are useful for quality monitoring into groups or regions. The triplets collected could be visualised to be those that would fall within selected regions of a three dimensional scatter plot of the values of the first, second and third signals. The regions could be drawn on to such a visualisation.
The first signal data sequence may be represented as the sequence D1,D2, . . . , Dxcex7xe2x88x921, Dxcex7, and the second signal data sequence represented as the sequence xcex931, xcex932, . . . , xcex93xcex7xe2x88x921, xcex93xcex7. The total number of data points xcex7 must be 2 or higher and a value of 1000 may be used. The artificial sequence numbered s is the sequence A1,s,A2,s, . . . , Axcex7xe2x88x921,s, Axcex7,s. The artificial sequence number s varies from 1 to a maximum value "sgr", "sgr" must be 1 or higher and a value of 5 may be used. The member n of the artificial sequence numbered s, An,s, may be determined from:                               A                      η            ⁢                          .              5                                      =                                            ∑                              κ                =                1                            η                        ⁢                          xe2x80x83                        ⁢                                          Ψ                ⁡                                  (                                      1                    ,                    κ                    ,                    n                    ,                    s                    ,                    t                                    )                                            ⁢                              D                κ                                              +                                    Ψ              ⁡                              (                                  2                  ,                  κ                  ,                  n                  ,                  s                  ,                  t                                )                                      ⁢                          Γ              κ                                                          (        1        )            
The coefficients "psgr" may depend on xcexa, the location of Dxcex7 in the first signal data sequence and also the location of xcex93xcex7 in the second signal data sequence: n, the location of An,s in the artificial data sequence numbered s; s, the artificial sequence number; and t, the time at which Dxcex7 and xcex93xcex7 were measured with respect to some specified time origin. The artificial signal generating means applies equation (1) repeatedly to calculate all values of An,s for n varying from 1 to xcex7, and s varying from 1 to "sgr". A useful choice for "psgr" is:
xcexa8(1,xcexa,n,s,t)=e(kxe2x88x92n)(T0xe2x88x92ST 1). . . (xcexaxe2x88x92n) less than 0
xcexa8(1,xcexa,n,s,t)=0 . . . (xcexaxe2x88x92n)xe2x89xa70
xcexa8(2,xcexa,n,s,t)="THgr" . . . xcexa=n
xcexa8(2,xcexa,n,s,t)=0 . . . xcexaxe2x89xa0nxe2x80x83xe2x80x83(2)
In equation (2) there is no explicit dependence on t. With this choice. equation (1) is close to a convolution of the first signal with a damped or decaying exponential added to the second signal multiplied by "THgr". The effective damping time constant is given by xcfx840+xcfx841s. The constants xcfx840 and xcfx841 set the range covered by the time constant as s varies from 1 to "sgr". The constant "THgr" sets the amount of second signal added.
The inclusion of an explicit dependence of "psgr" on time t or sequence number n is useful when the welding system properties are varying during the sampling for a signature. For example, in resistance spot welding, physical conditions vary substantially during one spot weld for which a single signature may be determined.
Grouping means form all possible sets of values of the type {Dn, s, An,s}, that is, sets consisting of a first signal data point Dn, an artificial sequence number s, and the corresponding member of the artificial sequence number s. An,s, n varies from 1 to xcex7, and s varies from 1 to "sgr". If there is only one artificial sequence, then s is always set to 1 in the sets of values.
Collection means collect sets of values which are useful for weld monitoring into groups or regions. The sets collected could be visualised to be those that would fall within selected volumetric regions of a three dimensional scatter plot with one axis plotting the value of the first signal, a second axis plotting the sequence number of the artificial sequence, and a third axis plotting the value of the corresponding artificial signal. If there is only one artificial sequence, all points will lie in the plane defined by s=1. The boundaries of the regions could be displayed as closed surfaces on such a visualisation.
The regions need not be of equal size, and they may be smaller where population density is greatest and may be exponentially greater in dimension, in both the first and artificial signal directions, as they progress away from the region of greatest population density. Once the regions are chosen, they are fixed during the weld monitoring process. The regions selected need not be contiguous, and regions may overlap.
Each of the collected sample points that fall within a given region are accumulated in the population of that region. The region populations can be represented by a population density function fr which is the population of the region numbered r, with r varying from 1 to xcfx81.
If a given point at {Dn, s, An,s} falls within region r, accumulation means increase the population fr by wr(Dn, xcex93n, An,s, n, s, t), where t is the time at which Dn and xcex93n were measured. wr(Dn, xcex93n, An,s, n, s, t) is the weight the point is given in region r. If wr is always one, for example, the populations are a simple count of the number of points in each region.
To produce the final adjusted region populations pr, function application means apply a single valued monotonic function F to each of the fr values:
pr=F(fr)xe2x80x83xe2x80x83(3)
for r=1 to xcfx81.
The complete set {p1, p2 . . . pxcfx81xe2x88x921, pxcfx81} of the pr collected is a single welding signature.
The weight functions wr are chosen to produce a welding signature which contains as much information about the properties of the final weld as possible for a given sampling rate and size. This may be done experimentally, by trial and error adjustment or by knowledge of the physical process. Since there is some statistical noise in the sample, it is useful to choose the wr to smooth the welding signature: this may be achieved by defining overlapping regions and decreasing wr for points closer to the boundary of region r. The function F is chosen to maximise the sensitivity of the welding signature to faults in the final weld.
The inclusion of a dependence of the weights wr on the data point number n permits windowing. Weights may be reduced near the start of the data sequence at n=1 and near the end of the sequence at n=xcex7 for example.
Generating a Combined Welding Signature
For a given process, it may be desirable to generate two single welding signatures using both the current and the voltage as the first signal. For processes such as tandem arc welding, two welding voltages and two currents can be measured. In this situation also, several single welding signatures may be generated.
When several single welding signatures are required, the sampling means collects several sequences of values for first signals and second signals. If the signals are related because they come from the same physical welding system, the signals may be sampled over the same time interval.
The apparatus for generating a single welding signature is then used to generate a single signature for each of the sequences of first and second signals, giving a total of xcexc single signatures. There are a total of xcfx81[m] regions defined for the single signature in. which contains the final adjusted region populations
{p1[m],p2[m],p3[m] . . . pp[m][m]}xe2x80x83xe2x80x83(4)
m varies from 1 to xcexc. Concatenation means then produce a combined welding signature PT by concatenating all the adjusted region populations in order from each single signature for m=1 to m=xcexc:
PT={p1[1], p2[1] . . . pxcfx81[1][1], p1[2], p2[2] . . . pxcfx81[xcexc][xcexc]}xe2x80x83xe2x80x83(5)
The number of elements xcfx81 in the combined welding signature is the sum of the elements in each single signature:                     ρ        =                              ∑                          m              =              1                        μ                    ⁢                      ρ            ⁡                          [              m              ]                                                          (        6        )            
The combined welding signature PT is manipulated in exactly the same way as are single signatures, and where signature is mentioned below, it refers to either a single or a combined signature.
Generating a Reference Welding Signature
The sampling means repetitively provides sequences of values and a new welding signature is produced for each sequence, or set of sequences in the case of combined signatures. Memory means retain a welding signature H={h1, h2 . . . hxcfx81xe2x88x921, hxcfx81}, or a set of welding signatures collected under welding conditions expected to be satisfactory and producing a high quality weldment. This may be reference data saved for some time, or could be data collected near the start of the welding run.
In the case of robotic welding, where a series of welds is carried out under conditions which may vary, either from weld to weld or during a single weld, a series of reference signatures may be stored and recalled when needed.
Reference signatures may also be calculated continuously during welding from previous sampling. In this case the reference is a weighted average of the N signatures H1, H2, H3 . . . HN where HN is the most recent signature determined. HNxe2x88x921 is the signature determined before that and so on. The reference signature X is determined from the weighted average
xr=W1h1r+W2h2r+W3h3r+ . . . +WNhNrxe2x80x83xe2x80x83(6xe2x80x2)
for r=1 to xcfx81, xr is the adjusted region population numbered r in the reference signature X: h1r to hNr are the adjusted region populations numbered r in the signatures H1 to HN determined from previous sampling; xcfx81 is the total number of regions in each signature: and W1 to WN are the signature weighting factors. The choice of signature weighting factors W1 to WN determines whether the reference X represents an average of weld signature behaviour over a relatively long period of time or represents recent welding behaviour.
The application of this approach to determining welding stability is described later.
Manipulation of Welding Signatures
Manipulation means manipulate welding signatures according to the requirements of the apparatuses described below.
When signatures are multiplied or divided by a number. it is understood that every adjusted region population in the signature should be multiplied or divided by the number to produce a new signature. Similarly when signatures are added or subtracted, the matching adjusted region populations in each signature are added or subtracted, that is, the adjusted region population numbered r in one signature is added or subtracted from the adjusted region population numbered r in the other signature for r=1, 2, 3 up to xcfx81.
The inner, or dot product, of any two signatures C and G is calculated as                               C          ·          G                =                              ∑                          r              =              1                        ρ                    ⁢                      xe2x80x83                    ⁢                                    C              r                        xc3x97                          g              r                                                          (        7        )            
where cr and gr are the adjusted populations of region r for signatures C and G respectively.
A normalised welding signature Cxe2x80x2 is calculated from a welding signature C as follows:
xe2x80x83Cxe2x80x2=C/({square root over (Cxc2x7C)})xe2x80x83xe2x80x83(8)
Estimation of the Probability of a Measured Signature Given a Set of Reference Signatures
Isotropic Reference Population
The probability of a measured signature S given a measured sample of xcex8 reference signatures R1, R2 to Rxcex8, is estimated by, firstly, using the signature manipulation means to calculate the average signature M:                     M        =                              (                                          ∑                                  J                  =                  1                                θ                            ⁢                              xe2x80x83                            ⁢                              R                J                                      )                    /          θ                                    (        9        )            
Secondly, the signature manipulation means are used to estimate the variance xcex62 based on distances between signatures:                               ζ          2                =                              [                                          ∑                                  J                  =                  1                                θ                            ⁢                                                (                                                            R                      J                                        -                    M                                    )                                ·                                  (                                                            R                      J                                        -                    M                                    )                                                      ]                    /                      (                          θ              -              1                        )                                              (        10        )            
In the special case when xcex8=1, and optionally for small values of xcex8, a value of xcex62 based on measurements made previously under similar welding conditions can be used.
Assuming a normalised population density distribution of signature distances from the average Z(y), the probability Π(S) of a signature being at the same distance as signature S or further from the average value within the reference signature population is:                               Π          ⁡                      (            S            )                          =                                            ∫                              |                                                      (                                          S                      -                      M                                        )                                    ·                                      (                                          S                      -                      M                                        )                                                  ⁢                                  |                  1                                ⁢                                  Z                  /                                                      v                    ⁢                                    Z              ⁡                              (                y                )                                      ⁢                          ⅆ              y                                                          (        11        )            
Finally, signature manipulation means evaluate the lower limit of the integral in equation (11), and statistical evaluation means evaluate Π(S).
Π(S) is a convenient measure of welding consistency, Π(S) is close to one for a measured signature at the average value and tends to zero for signatures far from the average value. Low values of Π(S) indicate that a fault has probably occurred in the process. A value of Π(S) less than 10xe2x88x924 may be used to indicate a fault in the process. Any simple statistical distribution can be chosen for Z(y). A useful choice is the Normal distribution:                               Z          ⁡                      (            y            )                          =                                            2              π                                ⁢                      e                                          -                                  1                  2                                            ⁢              2                                                          (        12        )            
where e is the base of the natural logarithms.
Statistical Distributions with Dimension Greater than One
Since signature images contain a number of variable elements, an improved method is to use a statistical distribution with dimension d greater than 1 and estimate the dimension d from the data.
Signature manipulation means are used to evaluate the average signature M and the variance xcex62 according to equations (9) and (10) respectively. Signature manipulation means are then used to evaluate the moment numbered J of the reference signature set xcexJ:                               λ          J                =                              [                                          ∑                                  J                  =                  1                                θ                            ⁢                                                (                                                            (                                                                        R                          J                                                -                        M                                            )                                        ·                                          (                                                                        R                          J                                                -                        M                                            )                                                        )                                J2                                      ]                    /                      (                          θ              -              1                        )                                              (                  10          xe2x80x2                )            
J should not be set to 2 since xcex2=xcex62 which has already been calculated, J=4 is a suitable choice.
A suitable choice for the normalised statistical distribution Z(y,d) with dimension d is a many-dimension Normal distribution:                               Z          ⁡                      (                          y              ,              d                        )                          =                                            2              ⁢                              d                d2                                                                    2                d2                            ⁢                              Γ                ⁡                                  (                                      d                    /                    2                                    )                                                              ⁢                      y                          d              -              1                                ⁢                      e                                          -                                  dy                  2                                            ⁢              2                                                          (                  11          xe2x80x2                )            
where xcex93 is the Gamma Function [1].
Using statistical evaluation means with J=4, d can be estimated from                     d        =                  2          /                      (                                                            λ                  4                                                  ζ                  4                                            -              1                        )                                              (                  12          xe2x80x2                )            
Once d is known. the probability Π(S) of a signature being at the same distance as signature S or further from the average value within the reference signature population is estimated from equation (11) with the many dimension population distribution Z(y,d) replacing Z(y):                               Π          ⁡                      (            S            )                          =                                            ∫                              |                                                      (                                          S                      -                      M                                        )                                    ·                                      (                                          S                      -                      M                                        )                                                  ⁢                                  |                  1                                ⁢                                  Z                  /                                                      v                    ⁢                                    Z              ⁡                              (                                  y                  ,                  d                                )                                      ⁢                          ⅆ              y                                                          (        13        )            
Π(S) is an improved measure of welding consistency.
Non Isotropic Reference Population
Although Π(S) of equations (11) or (13) is useful, the calculation assumes that the same distribution applies moving away in all directions in signature space from the average signature. In practice there may be correlations in the variations of elements of the signatures, and this information can be used to improve sensitivity to faults, since these will not necessarily exhibit such correlations.
To account for basic anisotropy, first it is necessary to estimate the direction in signature space M1 in which the reference sample shows the greatest deviations from the average M, that is M1 must maximise                               ζ          1          2                =                              (                                          ∑                                  J                  =                  1                                θ                            ⁢                              xe2x80x83                            ⁢                                                (                                                            (                                                                        R                          J                                                -                        M                                            )                                        ·                                          M                      1                                                        )                                2                                      )                    /                      [                                          (                                  θ                  -                  1                                )                            ⁢                              (                                                      M                    1                                    ·                                      M                    1                                                  )                                      ]                                              (        14        )            
for the measured sample of xcex8 reference signatures R1, R2 to Rxcex8, with average signature M, xcex8 must be greater than 2.
Direction location means are used to estimate M1. For a set of linearly independent reference samples, these may specify M1 as a linear combination:                               M          1                =                              R            1                    -          M          +                                    ∑                              J                =                2                            θ                        ⁢                                          ϕ                J                            ⁡                              (                                                      R                    J                                    -                  M                                )                                                                        (        15        )            
The unknown coefficients xcfx862, xcfx862 to xcfx86xcex8 can then be found by using a numerical method to minimise 1/xcex612. A conjugate gradient method such as the Polak-Ribiere method may be used.
Once M1 is known, it is normalised to give the unit signature M1xe2x80x2 according to equation (8).
Differences from the average are now split into components parallel and perpendicular to M1xe2x80x2 and treated independently statistically. Because of the definition of the overall average M, necessarily the average of the components of the sample differences along M1xe2x80x2 will equal zero.
Signature manipulation means are used to find the variance of the components parallel to M1xe2x80x2, xcex612:                               ζ          1          2                =                              (                                          ∑                                  J                  =                  1                                θ                            ⁢                              xe2x80x83                            ⁢                                                [                                                            (                                                                        R                          J                                                -                        M                                            )                                        ·                                          M                      1                      xe2x80x2                                                        ]                                2                                      )                    /                      (                          θ              -              1                        )                                              (        16        )            
Signature manipulation means are used to find the remainder of Rjxe2x88x92M which is orthogonal to M1xe2x80x2 for each reference signature from j=1 to j=xcex8
Rj[1]=Rjxe2x88x92Mxe2x88x92((Rjxe2x88x92M)xc2x7M1xe2x80x2)M1xe2x80x2xe2x80x83xe2x80x83(17)
Signature manipulation means are then used to find the variance of these orthogonal remainders (xcex6[1])2 according to:                                           (                          ζ                              (                1                )                                      )                    2                =                              [                                          ∑                                  J                  =                  1                                θ                            ⁢                              xe2x80x83                            ⁢                                                R                  J                                      (                    1                    )                                                  ·                                  R                  J                                      (                    1                    )                                                                        ]                    /                      (                          θ              -              1                        )                                              (        18        )            
Assuming independent distributions, the combined probability Π(S) that the component of a signature""s difference from the average along M1xe2x80x2 is greater than or equal to |(Sxe2x88x92M)xc2x7M1xe2x80x2|, and the signature""s remainder orthogonal to M1xe2x80x2 is greater than or equal to the remainder of S, is given by the product of the individual probabilities (assumed independent):                               Π          ⁡                      (            S            )                          =                              ∫                                                            (                                      |                                                                  S                                                  (                          1                          )                                                                    ·                                              S                                                  (                          1                          )                                                                                      |                                    )                                1.2                            /                              z                                  (                  1                  )                                                      v                    ⁢                                    Z              ⁡                              (                y                )                                      ⁢                          ⅆ              y                        ⁢                          xe2x80x83                        ⁢                                          ∫                                  |                                      (                                          S                      -                      M                                        )                                    |                                                            ·                                              M                        1                                                              ⁢                                          γz                      1                                                                      v                            ⁢                                                Z                  ⁡                                      (                    y                    )                                                  ⁢                                  ⅆ                  y                                                                                        (        19        )            
where the remainder
S[1]=Sxe2x88x92Mxe2x88x92((Sxe2x88x92M)xc2x7M1xe2x80x2)M1xe2x80x2xe2x80x83xe2x80x83(20)
When anisotropies are taken into account, the previous apparatus for the isotropic case is used up to equation (9) for determining the average signature. Then the anisotropic apparatus is employed to evaluate from equation (14) to equation (18). Finally, signature manipulation means calculate the lower limits of the integrals in equation (19) using equation (20) and statistical evaluation means evaluate Π(S) from equation (19).
Π(S) is used as previously described for isotropic reference signature distributions.
For the distribution of remainders orthogonal to Mxe2x80x21, it is an improvement to use a many dimension distribution Z(y,d[1]) so that equation (19) is replaced with                               Π          ⁡                      (            S            )                          =                                            ∫                                                                    (                                          |                                                                        S                                                      (                            1                            )                                                                          ·                                                  S                                                      (                            1                            )                                                                                              |                                        )                                                        1.2                    ⁢                                          xe2x80x83                                                                      ⁢                                  z                                      (                    1                    )                                                                        v                    ⁢                                    Z              ⁡                              (                                  y                  ,                                      d                                          (                      1                      )                                                                      )                                      ⁢                          ⅆ              y                        ⁢                          xe2x80x83                        ⁢                                          ∫                                                                            |                                              (                                                  S                          -                          M                                                )                                            |                                              ·                                                  M                          1                                                                                      "RightBracketingBar"                                    ⁢                                      z                    1                                                  v                            ⁢                                                Z                  ⁡                                      (                    y                    )                                                  ⁢                                  ⅆ                  y                                                                                        (        21        )            
Using statistical evaluation means for a many-dimension Normal distribution [equation (11xe2x80x2)], the dimension d[1] is estimated from                                           d                          (              1              )                                =                      2            /                          (                                                                    λ                    4                                          (                      1                      )                                                                                                  (                                              ζ                                                  (                          1                          )                                                                    )                                        4                                                  -                1                            )                                      ⁢                  
                ⁢        where                            (        22        )                                          λ          4                      (            1            )                          =                              [                                          ∑                                  J                  =                  1                                θ                            ⁢                                                (                                                            R                      J                                              (                        1                        )                                                              ·                                          R                      J                                              (                        1                        )                                                                              )                                2                                      ]                    /                      (                          θ              -              1                        )                                              (        23        )            
Once d[1] is known, statistical evaluation means are used to evaluate Π(S) from equation (21), which is then used as described previously.
Apparatus for Signature Conversion to Basis Set Description
If a large number of reference signatures are used for welding consistency determination and fault detection, several potential difficulties arise:
(1) Storage of signatures in computer memory or on hard disk may become an issue
(2) Calculation of inner products according to equation (7) may consume a considerable amount of time. For example, to find the direction in signature space M1 in which the reference sample shows the greatest deviations from the average requires calculation of several inner products.
(3) Linear independence of the signatures cannot be guaranteed.
To overcome these difficulties, signatures may successively be represented by their coordinates in an appropriate subspace with the following modified Gram-Schmidt orthogonalization apparatus.
Suppose that a set R1, R2 . . . Rxcex8 of signatures is to be sequentially converted to a subspace description, the jth signature R1 is the next to be converted, and an orthonormal basis set B1xe2x80x2, B2xe2x80x2 . . . B"xgr"xe2x80x2 has already been generated by the apparatus. Orthonormal implies that all pairs of basis signatures are orthogonal, so that:
Bxe2x80x2kxc2x7Bxe2x80x2k1=0xe2x80x83xe2x80x83(24)
for k=k1 and k1 and k in the range 1 to "xgr", and each value is normalised according to equation (8), so that
Bxe2x80x2kxc2x7Bxe2x80x2k=1xe2x80x83xe2x80x83(25)
for k in the range 1 to "xgr".
Signature manipulation means evaluate the next prospective basis component coming from Rj as                               B                      ξ            -            1                          =                              R            J                    -                                    ∑                              k                =                1                            ξ                        ⁢                          xe2x80x83                        ⁢                                          (                                                      R                    J                                    ·                                      B                    k                                                  )                            ⁢                              B                k                xe2x80x2                                                                        (        26        )            
However basis addition means only accept B"xgr"+1 for the basis set if it satisfies                                           B                          ξ              -              1                                ·                      B                          ξ              -              1                                      ≻                                            ϵ              2                        ⁡                          (                                                ∑                                      Ji                    =                    1                                    J                                ⁢                                                      R                    Ji                                    ·                                      R                    Ji                                                              )                                /          J                                    (        27        )            
The value of xcex5 is chosen such that B"xgr"+1 will only be accepted for the basis set if it is sufficiently large. This means that if Rj is not significantly linearly independent of the existing basis it will not be used to generate a new basis set signature. A value of xcex5 which may be used is 0.001.
If B"xgr"+1 is to be accepted, the normalised value B"xgr"+1xe2x80x2 is included in the basis set, applying equation (8).
To start the process off, basis addition means always include the normalised first reference R1xe2x80x2 as the first element of the basis set.
Finally, basis description means express R1 in terms of basis set coordinates xcex1j1, xcex1j2, . . . xcex1j"Xgr" in the subspace defined by the set B1xe2x80x2, B2xe2x80x2 . . . B"Xgr"xe2x80x2, where the set extends to "Xgr"="xgr"+1 if B"xgr"+1 was accepted and to "Xgr"="xgr" otherwise:                               R          j                =                              ∑                          k              =              1                        Ξ                    ⁢                                    a              jk                        ⁢                          B              k              xe2x80x2                                                          (        28        )            
where
xcex1jk=Rjxc2x7Bxe2x80x2kxe2x80x83xe2x80x83(29)
When basis addition means complete the process through to Rxcex8, the description of the reference signatures is made up of a set of xcexa9 basis signatures B1xe2x80x2, B2xe2x80x2 . . . Bxcexa9xe2x80x2 plus a set of xcex8 basis set coordinates with xcexa9 values in each. If a particular basis signature does not appear in the expression for a given reference because the given basis signature was added later, the corresponding coordinate is set to zero.
This will generally be a more compact description of the welding signatures when xcex8 is greater than about 10. For real time work, basis addition means are used with consecutive reference signatures as they are determined during welding.
Another advantage of using the basis set description is that signature manipulation can be performed more efficiently. Basis coordinate manipulation means perform the operations. Suppose C and G are signatures expressed in the basis set corresponding to the references R1 to Rxcex8 as                               C          =                                    ∑                              k                =                1                            Ω                        ⁢                                          γ                k                            ⁢                              B                k                                                    ⁢                  
                ⁢                  G          =                                    ∑                              k                =                1                            Ω                        ⁢                                          δ                k                            ⁢                              B                k                xe2x80x2                                                                        (        30        )            
where the basis set coordinates are
xcex3k=Cxc2x7Bxe2x80x2k
xcex4k=Gxc2x7Bxe2x80x2kxe2x80x83xe2x80x83(31)
for k=1 to k=xcexa9. The inner product is given by                               C          ·          G                =                              ∑                          k              =              1                        Ω                    ⁢                                    γ              k                        ⁢                          δ              k                                                          (        32        )            
This operation only requires xcexa9 multiplications as compared to the xcfx81 multiplications for signatures.
When signatures expressed in terms of the basis set are multiplied or divided by a number, every basis set coordinate should be multiplied or divided by the number to produce a new signature expressed in terms of the basis set. Similarly when signatures are added or subtracted, the matching coordinates for each signature are added or subtracted respectively, that is, the coordinate numbered k in one set is added or subtracted from the coordinate numbered k in the other set for k=1, 2, 3 up to xcexa9.
Basis set descriptions cannot be applied to every measured signature. They only apply to signatures that lie in the signature subspace defined by the basis set of signatures. These signatures are the reference signatures themselves and linear combinations of the reference signatures.
Determining Welding Stability
During a welding process, a quantitative measure of what is loosely called xe2x80x9cstabilityxe2x80x9d can be very valuable. A useful measure is the variability of the welding signature, that is, is a given signature as one would expect it from the previous signatures? Suppose a sequence H1, H2 . . . HN of signatures has been determined by signature generation means, with N greater than 1. Signature manipulation means determine a simple linear prediction of the next signature X using a particular choice of signature weighting factors [equation (6xe2x80x2)]:
X=2HNxe2x88x92HNxe2x88x921xe2x80x83xe2x80x83(33)
The squared distance between the next measured signature SNxe2x88x921 and the predicted signature is
(SNxe2x88x921xe2x88x92X)xc2x7(SNxe2x88x921xe2x88x92X)xe2x80x83xe2x80x83(34)
Assuming a signature population distribution Z(y), such as the Normal distribution [equation (12)], the probability of the signature SN+1 is calculated using statistical evaluation means from                               Π          ⁡                      (                          S                              N                +                1                                      )                          =                              ∫                                                            (                                                            (                                                                        S                                                      N                            -                            1                                                                          -                        X                                            )                                        ·                                          (                                                                        S                                                      N                            -                            1                                                                          -                        X                                            )                                                        )                                                  1                  /                  2                                            /                                      ∞                    ⁢                                    Z              ⁡                              (                y                )                                      ⁢                          ⅆ              y                                                          (        35        )            
with xcex6 given a fixed value based on a reference set of signatures previously generated, using equation (10). Π(SN+1) is a useful stability measure with higher probabilities indicating a more stable welding process.
If the welding process is relatively unstable. a coarser measure of stability comes from considering a larger group of signatures immediately prior to the measured signature SN+1 as a reference set R1 to Rxcex8 and using statistical evaluation means to determine Π(SNxe2x88x921) from equation (11), equation (19), or equation (21)
Use of Invention
Use of the invention may allow determination of whether a fault has occurred in a welding process, while the process is under way. It may also allow determination whether a welding process is consistent and stable, while the process is under way.
In further aspects the invention, as currently envisaged. is methods of performing the steps of the processes being performed above, but without reference to any specific apparatus.